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Longest Cycles in 3-Connected 3-Regular Graphs

Published online by Cambridge University Press:  20 November 2018

J. A. Bondy  and
M. Simonovits
J. A. Bondy
Affiliation:
University of Waterloo, Waterloo, Ontario
M. Simonovits
Affiliation:
Eötvös Loránd University, Budapest, Hungary
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In this paper, we study the following question: How long a cycle must there be in a 3-connected 3-regular graph on n vertices? For planar graphs this question goes back to Tait [6], who conjectured that any planar 3-connected 3-regular graph is hamiltonian. Tutte [7] disproved this conjecture by finding a counterexample on 46 vertices. Using Tutte's example, Grunbaum and Motzkin [3] constructed an infinite family of 3-connected 3-regular planar graphs such that the length of a longest cycle in each member of the family is at most nc, where c = 1 – 2–17 and n is the number of vertices. The exponent c was subsequently reduced by Walther [8, 9] and by Grùnbaum and Walther [4].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 4 , April 1980 , pp. 987 - 992
Copyright
Copyright © Canadian Mathematical Society 1980

References

1. Barnette, D., Trees in polyhedral graphs, Can. J. Math. 18 (1966), 731–736.Google Scholar
2. Bondy, J. A. and Entringer, R. C., Longest cycles in 2-connected graphs with prescribed maximum degree, Can. J. Math., to appear.CrossRefGoogle Scholar
3. Grünbaum, B. and Motzkin, T. S., Longest simple paths in polyhedral graphs, J. London Math. Soc. 37 (1962), 152–160.Google Scholar
4. Grünbaum, B. and Walther, H., Shortness exponents of families of graphs, J. Combinatorial Theory Ser. A. 14 (1973), 364–385.Google Scholar
5. Lang, R. and Walther, H., Über längste Kreise in reguldren Graphen, Beitrage zur Graphentheorie (Kolloquium, Manebach, 1967). Teubner, Leipzig (1968), 91–98.Google Scholar
6. Tait, P. G., Remarks on colouring of maps, Proc. Royal Soc. Edinburgh Ser. A. 10 (1880), 729.Google Scholar
7. Tutte, W. T., On hamiltoman circuits, J. London Math. Soc. 21 (1946), 98–101.Google Scholar
8. Walther, H., Über die Anzahl der Knotenpunkte eines längsten Kreises in planaren, kubischen dreifach knotenzusammenhdngenden Graphen, Studia Sci. Math. Hungar. 2 (1967), 391–398.Google Scholar
9. Walther, H., Über Extremalkreise in reguldren Landkarten, Wiss. Z. Techn. Hochsch. Ilmenau 15 (1969), 139–142.Google Scholar